A matt diffuser plate of radiance W m sr sits in air ( )

PHYSICS · Multi-part · Fermi University Physics · Optics & Photonics · Geometrical optics and imaging · Geometrical optics and imaging

Optics and Photonics, Ch Radiometry and photometry

Problem

A matt diffuser plate of radiance L1=2.0×104L_1 = 2.0\times10^{4} W m−2^{-2} sr−1^{-1} sits in air (n1=1n_1 = 1), pressed against the flat, polished end of a glass rod (n2=1.50n_2 = 1.50). Follow a narrow beam that leaves a patch of the plate of area A=1.0A = 1.0 mm2^2 into a cone of half-angle u1=0.10u_1 = 0.10 rad about the normal and crosses the flat interface into the glass. Treat the interface as lossless: nothing is absorbed, scattered or turned back at it, and the beam's cross-section AA is the same on both sides. ((a)) Using the paraxial Snell's law n1u1=n2u2n_1 u_1 = n_2 u_2, find the half-angle u2u_2 of the cone inside the glass and the solid angles Ω1\Omega_1 and Ω2\Omega_2 of the cone on the two sides. ((b)) Find the flux Φ\Phi the beam carries, and show numerically that the etendue G=n2AΩG = n^2 A\Omega is the same on both sides of the interface. ((c)) The flux and the cross-section are unchanged by the interface, so find the radiance L2L_2 of the beam inside the glass from Φ=L2AΩ2\Phi = L_2 A\Omega_2, and check that L2/n22=L1/n12L_2/n_2^2 = L_1/n_1^2. ((d)) At the far end the rod is polished flat and the beam leaves into air again. What radiance does it leave with? Has the glass made the light any brighter?

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